| L(s) = 1 | + (−12.2 + 12.2i)2-s + (41.5 − 100. i)3-s + 209. i·4-s + (−793. − 328. i)5-s + (722. + 1.74e3i)6-s + (7.18e3 − 2.97e3i)7-s + (−8.87e3 − 8.87e3i)8-s + (5.58e3 + 5.58e3i)9-s + (1.37e4 − 5.71e3i)10-s + (3.60e4 + 8.69e4i)11-s + (2.10e4 + 8.70e3i)12-s + 6.90e4i·13-s + (−5.17e4 + 1.24e5i)14-s + (−6.59e4 + 6.59e4i)15-s + 1.10e5·16-s + (2.68e5 + 2.15e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.543 + 0.543i)2-s + (0.296 − 0.714i)3-s + 0.409i·4-s + (−0.567 − 0.235i)5-s + (0.227 + 0.549i)6-s + (1.13 − 0.468i)7-s + (−0.765 − 0.765i)8-s + (0.283 + 0.283i)9-s + (0.436 − 0.180i)10-s + (0.741 + 1.79i)11-s + (0.292 + 0.121i)12-s + 0.670i·13-s + (−0.359 + 0.869i)14-s + (−0.336 + 0.336i)15-s + 0.422·16-s + (0.780 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.27584 + 0.672966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27584 + 0.672966i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.68e5 - 2.15e5i)T \) |
| good | 2 | \( 1 + (12.2 - 12.2i)T - 512iT^{2} \) |
| 3 | \( 1 + (-41.5 + 100. i)T + (-1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (793. + 328. i)T + (1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-7.18e3 + 2.97e3i)T + (2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-3.60e4 - 8.69e4i)T + (-1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 6.90e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-5.67e5 + 5.67e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (-6.36e5 - 1.53e6i)T + (-1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (4.94e6 + 2.04e6i)T + (1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (1.49e6 - 3.60e6i)T + (-1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-4.00e6 + 9.66e6i)T + (-9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-1.02e7 + 4.24e6i)T + (2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-8.66e6 - 8.66e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 - 1.96e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (5.00e7 - 5.00e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (3.24e7 + 3.24e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (-5.98e7 + 2.48e7i)T + (8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 - 5.04e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (1.12e8 - 2.72e8i)T + (-3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (6.12e7 + 2.53e7i)T + (4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (1.82e8 + 4.39e8i)T + (-8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-2.01e8 + 2.01e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 7.06e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (8.43e8 + 3.49e8i)T + (5.37e17 + 5.37e17i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.23388366904617707722695088153, −15.75420004045351921400977876702, −14.40296434513090175218181304800, −12.77812920234007002540033219534, −11.64851310619719566835337873760, −9.370552214553931637842739937447, −7.67325159029368953231998553163, −7.29734694721541381988502108605, −4.30500084229722911273910581672, −1.51910239792167682432047033339,
1.00352246093842481045596667338, 3.36712421560352819648331192614, 5.52872213036317067273615879949, 8.206884307052581363139336945840, 9.434404969910010733845883241262, 10.91368417419565567819436297384, 11.77407794756486299279320085438, 14.31835211480933680285762097634, 15.03194125088351630258893663580, 16.47065933788839257457292329887
